Basic Thermodynamics of the Atmosphere

We consider a model for atmospheric circulation based on the Euler equations for a compressible gas. We consider two hydrostatic base solutions depending on height, one with constant temperature and one isentropic with constant temperature gradient or lapse rate. We argue that these solutions represent solutions with maximal and minimal turbulent dissipation with the observed real lapse rate soemwhere in between. We find an atmospheric cyclic thermodynamic process of rising-expanding-cooling and descending-compressing-warming air, which is similar to that of an air conditioner or refrigerator. We seek solutions as perturbations of the hydrostatic base solutions with the perturbations satisfying a modified form of the incompressible Euler equations. 1 Compressible/Incompressible Euler as Climate Model As a model of the atmosphere we consider the Euler equations for a compressible prefect gas occupying a volume Ω: Find (ρ, u, T ) with ρ density, u velocity and T temperature depending on x and t > 0, such that for x ∈ Ω and t > 0: Duρ+ ρ∇ · u = 0 Dum+m∇ · u+∇p+ gρe3 = 0 DuT +RT∇ · u = q (1) where m = ρu is momentum, p = RρT is pressure R = cp−cv with cv and cp specific heats under constant volume and pressure, and Duv = v̇ + u · ∇v is the material time derivative with respect to the velocity u with v̇ = ∂v ∂t the partial derivative with respect to time t, e3 = (0, 0, 1) is the upward direction, g gravitational accelleration and q is a heat source. For air cp = 1 and cp cv = 1.4. The Euler equations are complemented by initial values for ρ, m and T at t = 0, and the boundary condition u · n = 0 on the boundary of Ω where n is normal to the boundary. Computer Science and Communication, KTH, SE-10044 Stockholm, Sweden.